Hyperelliptic Curves

where $h$ is a polynomial with $\deg h \le g$, and $f$ is a monic
polynomial with $\deg f \le 2g + 1$. Elliptic curves satisfy this
definition for $g = 1$.

As for elliptic curves, each hyperelliptic curve contains a single
point at infinity which we denote $O$.
For a point $P = (x, y)$ on a hyperelliptic curve,
let $\tilde P$ be the point $(x,-y - h(x))$. (So for elliptic curves,
$\tilde P = -P$.) If $P = \tilde P$ then call $P$ a special point,
otherwise call it ordinary. (On elliptic curves, special points are
points of order 2.)

However, in general there is no group structure on the set of points
of a hyperelliptic curve. Instead, we work on the jacobian group, which
is defined below.

The Jacobian Group

Note that the definitions of function fields and divisors apply
to any curve $C$.
We define the jacobian group to be
$Div^0(C) / Prin(C)$.

It turns out that each element of the jacobian group is equivalent
to a divisor of the form

\[
\sum m_i P_i - (\sum m_i ) O
\]

where
$m_P \ne 0$ implies $m_{\tilde P} = 0$ unless $P$ is special in which
case $m_P = 1$, and $\sum m_i \le g$.
For elliptic curves, each element of the jacobian is equivalent to
$P - O$ for some point $P$, and addition on points induces addition
in the jacobian.